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The table shows the values of two functions, and , where is the population of the US, in millions, years after 2010, and is the number of books sold per year, in millions, years after 2010.
| (years since 2010) | (millions) | (millions) | |
|---|---|---|---|
| 0 | 2,530 | 309.35 | |
| 1 | 2,400 | 311.64 | |
| 2 | 2,730 | 313.99 | |
| 3 | 2,720 | 316.23 | |
| 4 | 2,700 | 318.62 | |
| 5 | 2,710 | 321.04 | |
| 6 | 2,700 | 323.41 |
Plot the values of as a function of . What does the plot tell you about book sales?
Here are the graphs of two functions, and . Define a new function by adding and , so . On the same axes, sketch what you think the graph of looks like.
We can add, subtract, multiply, and divide functions to get new functions. For example, the cost in dollars of producing cups of lemonade at a lemonade stand is . The revenue (amount of money collected) from selling cups is dollars. The profit from selling cups is the revenue minus the cost, so
Here are the graphs of , , and . Can you see how each value on is the result of the difference between the corresponding points on and ?
The average profit per cup, , from selling cups, is the quotient of the profit and the number of cups, so
Graph of three lines, origin O. Horizontal axis from 0 to 10 by 1's, labeled number of cups. Vertical axis from negative 8 to 20 by 2's, labeled dollars. Line C starts at 0 comma 5 passes through 5 comma 4 and ends at 10 comma 13. Line R starts at 0 comma 0, passes through 5 comma 8, and ends at 10 comma 20. Line P starts at negative 5 comma 0, passes through 5 comma 1, and ends at 10 comma 5.
Here are the graphs of and . Can you see how the value of is the result of the quotient of and ? Why does it make sense that both functions are negative when and positive when ?
Graph of two lines, origin O. Horizontal axis from 0 to 10 by 1's, labeled number of cups. Vertical axis from negative 8 to 20 by 2's, labeled dollars. Line P starts at negative 5 comma 0, passes through 5 comma 1, and ends at 10 comma 5. Line A starts near 0 point 5 comma negative 8, curves upwards and to the right passing through 1 comma negative 3 point 8, 4 comma negative 0 point 05, 5 comma 0 point 2, ending at 10 comma 0 point 7.
Since can only be positive, and always have the same sign for a given value. Notice that for the average profit to be positive, the seller has to sell at least 5 cups (since is not in the domain, we must round up). It is also true that for a large number of cups, the average profit is close to \$1.20 per cup.